Rationality of Strategies in Games Theory Shichang Zhang

Stat 157

Professor Aldous

April 20 2016

Abstract

This is an extended write-up of analyzing a well-known game interested both psychologists and

game theorists called “guess 2/3 of the average”; all the players choose an integer from 1-100

simultaneously; whose choice is the closest to 2/3 of the average of all these numbers will win and

get 50 dollars as a prize. This paper also analyses two variations of the original game. One variation

changes the way of calculating the winning number, and the other changes the prize amount. The

analysis focuses on two perspectives. One is breakdown the game and analyze different types of

strategies on the psychological side to illustrate different levels of rationality of these strategies.

The other one is trying to come up with an optimal strategy that will win the game. I build a simple

model for the optimal strategy using it to simulate data and compare the result with a larger dataset

from the previous experiment with New York Times readers.

1.   Introduction

The game “Guess 2/3 of the Average” was created by Alain Ledoux in 1981. He used this

game as a tie breaker to select the winner among 4000 magazine readers, who reached the same

amount of points in the previous puzzle. Then Rosemarie Nagel revealed that this kind of games

can be used to disclose participants’ “depth of reasoning”. Also, due to the analogy of Keynes’s

comparison of newspaper beauty contests and stock market investments, the guessing game is

known as the Keynesian beauty contest. Now, the beauty contest game becomes famous in

experimental economics to illustrate the difference of perfect rationality and the common

knowledge of rationality.

In this paper, I analyze one of the most widely-spread versions of “Guess 2/3 of the Average”,

which asks players to choose integers from 1 to 100, and provide the reason why they want to

make these choice. Also, two variations of this classic game are included. In the analysis of these

three versions of the game, I breakdown and categorize all the players’ strategies to illustrate

different levels of rationality of these strategies. There is also a model, which is a combination of

all these strategies, elucidates the optimal strategy. The final step in this study is comparing the

simulated data of the optimal strategy using R with previous data from the New York Times

readers.

2.   Background and Game Description

38 people ranging from age 18 to age 56 and forming a variety of educational backgrounds

were invited to play this game. Besides the original game, players were also asked for two more

questions which could be considered as variations of the primary game. As a result, each person

played the game three times with slightly different rules each time, also their answers and strategies

of making their choices were recorded.

For the original game, all the participants in this game choose an integer from 1-100

simultaneously. Then the average of these numbers will be calculated. Whose choice is the closest

to 2/3 of the average will be the winner and get 50 dollars as a prize.

For variation 1, the way of calculating the average is different. After the data has been collected,

the highest 5% choices of numbers will be considered outliers. In other words, the biggest 5%

choices will be removed first, and only the left 95% will be used to do the subsequent calculation,

so the mean will be a trimmed mean. The rest of the game remains the same.

For variation 2, almost all the rules are the same with the original game (not variation 1). The

only difference is that the prize amount. The winner will get $x rather than $50, where x is the

number that the winner chooses.

3.   Data

All the players choose 3 integers from 1-100. In the first plot (top left), every 3 numbers on

one vertical line are the numbers chosen by the same person, which shows how people change

their answers in different scenarios. Black numbers are answers of the original game, and blue

numbers for variation 1, and red numbers for variation 2. The data is sorted increasingly based on

answers of the original game. The other three plots are the separate data plots for each game, which

indicate the range and distribution of each game more clearly.

4.   Analysis of the Original Game

4.1   Categorizing

All the players were asked for the reason why they chose that number, and players with similar

strategies were put together as a group. Overall, there are 5 different groups based on the players’

strategies.

•   Naive players

•   Common-rational players I

•   Common-rational players II

•   Hyper-rational players

•   The best players

In the following analysis for each group, I will show the recorded information including

common answers for the players in the same group, their common responses, and how many

players are assigned to this group. Also I will analyze their strategies and state a method to simulate

similar answers using all these information.

4.2   Native players

Generally, native players can choose any number as their answers, because they are simply

guessing. Their common responses can be “I am just guessing”, or “I feel like this number is right”.

Out of 38, 14 people said they guessed the answer.

Naive players in fact have no strategy, and they choose the number randomly. These players

usually do not read the game description carefully. Sometimes, although they read, they do not

quite understand what the problem is asking. Most of the time, naïve players just pick numbers

they like or they feel is right, so I try to use uniform [1,100] to simulate these kind of answers.

Then further analysis of native players shows that the result should not be this simple. First, if

a player spends a few minutes to think about the question, it is easy to tell that the highest winning

number can only be 67, which happens when everyone chooses 100. Therefore, I was expecting

that no one would chose numbers greater than 67 before I saw the data, but the highest answer was

70 as the following plot shows. It means that naïve players did not realize this situation.

However, besides the highest number, another occurrence is obvious in this plot. Among all

the choices of 14 naive players, the lowest answer can be as small as 3, but no one chose any

number bigger than 70 even the range is 1 to 100. This shows that even naïve players said they

were purely guessing, uniform [1,100] may not be a good approximation. Hence, I asked these

players if they had thought about 100 ×  2/3 is 67. Some of them simply said no, but some said that

they felt like they shouldn’t choose a number too big, because they saw a 2/3 and also the word

average in the description. Although they could not clearly explain why these words should

influence their answers, these players did not purely guess. At least they got some information

from the description. Therefore, uniformly on [1, 67] should work better than uniform on [1,100]

for those players who were influenced by the word “average” and “2/3”. I modified my simulation

to be uniformly distributed on [1,67] for these players, and uniformly on [1,100] for others.

4.3   Common-rational players I

One of the most common strategies leads players to choose numbers like 33 or 34. When

talking about their reasons for making these choices, they always say something like, “from 1 to

100, the average should be around 50, and 2/3 of it is 33”. Out of all 38 players, 12 players used

this strategy.

These players use a simple strategy. Their strategy is based on an assumption that other answers

are uniform and the average is around 50, so they will win if they choose 2/3 of 50. Since some

players who chose answers like 30 also gave similar reasons, these answers are not exactly uniform.

I apply Normal (33.5, 1) to simulate answers of using this strategy.

One problem about this strategy is that the assumption, which is others choose numbers

uniformly on [1,100], only happens when all other players have no specific strategies.

Unfortunately, this assumption is not true. Even the naïve players in this experiment are not all

choosing numbers uniformly. For these common-rational players, if their opponents are random

number generators, to be more rigorous a large number of random number generators, their

strategy will be a good one. In practice, players who use this strategy, oversimplify the game.

4.4   Common-rational players II

Another common strategy is used by players who choose 22 or 23 as their answers. The typical

reason is like, “numbers bigger than 67 are unlikely to win, so the problem is actually asking for

choosing numbers on [1, 67] rather than [1, 100]. The average on [1, 67] is 34, and 2/3 of the

average is 22”. This sounds reasonable, and 5 players used this strategy.

These players realize that 2/3 of 100 cannot be bigger than 67. It is certainly right, because

even if all the players in this game choose 100, the final answer can only be as big as 67. Normally,

a player should not expect all the other people to choose 100, unless this player can control others’

mind. Then the answer can be any number the player wants it to be. The player should not worry

about choosing numbers. Actually, there is one circumstance for the winning number to be greater

than 67. It happens when all the answers are greater than 67, and the smallest one among them will

win. In this case, because only the smallest number can win, all the players want to keep reducing

their answers to be the smallest one, and they will reach numbers under 67 eventually. The

conclusion is that answers greater than 67 are not plausible to win. In game theory, we say that

these numbers are dominated.

In this case, rational choices should be in the region [1, 67]. Assume players choose uniformly

on [1, 67]. Then the average should be around 34, and 2/3 of the average should be around 22.5.

Similar to the last one, Normal (22.5, 1) is used to simulate these answers.

Players who use this strategy are making the same mistake as pervious common-rational

players. After they have done their first step analysis, they assume that all the other answers are

uniformly distributed on [1, 67]. Because there are many native players who do not realize the

domination, this assumption certainly cannot be guaranteed. Besides native players, players use

this strategy also underestimate some hyper-rational players who think through more steps.

4.5   Hyper-rational players

In many different versions of these guess average games, there are always players choose the

smallest choice as their answer. Here it is 1. Their reasons are usually like this, “first, all the

numbers bigger than 2/3 of 100 are dominated, so we only need to look at the numbers on [1, 66].

Then, do the same analysis for these numbers. It leads to numbers on [1, 44]. Keep doing this again

and again. The answer will end up with 1”. Comparing to all the previous answers, their strategy

sounds pretty reasonable, and 3 out of 38 players gave this kind of reason and the answer 1.

Hyper-rational players can also be called perfect-rational players, but it does not mean their

answers are perfect. It means that these players do perfect-rational analysis only tenable in theory.

These players are smart, and also some of them have learnt game theory before. They solve this

problem by first assuming that all the players are hyper-rational, which is exactly the assumption

people usually make when doing a game theory problem. After that, hyper-rational players do

much further analysis than those 2 different kinds of common-rational players mentioned before.

In the situation where all the players are hyper-rational, they would all realize that the numbers

greater than 67 are dominated. This means now they are playing a new game which is choosing a

number from 1 to 67. Because hyper-rational players know that all the players should get this idea,

they can go one more step further. That is all the numbers greater than 2/3 of 67 are also dominated.

Then the game becomes choosing a number from 1 to 45. All the players do the similarly analysis

again and again. Each time the upper limit of the valid interval goes down by 1/3 of it, until they

get the hyper-rational answer, 1. In game theory, we call this answer as a Nash Equilibrium. For

these players, their answers have no potential to deviate, so no complicated simulation is needed.

Simply the number 1 is used to simulate the answers for hyper-rational players.

However, hyper-rational players forget that this is not an ideal situation that everyone in this

game is hyper-rational. Not all the players can think through these many steps. The naive players

and common-rational players choose numbers much bigger than one. Their choices push the

average up, so the final answer is not plausible to be 1.

I asked the hyper-rational players if they had ever thought about that even though their logical

inference was totally right in theory, choosing 1 was still unlikely to win. In fact, they did know

that, but they told me 1 is the “correct” answer for this problem. Whether they will win is not

important. What important is their inference is perfect. I then realized that they chose 1, just

because they wanted to prove that they understand the game, and they are rational.

Hyper-rational players give good analysis and answers. However, they are still not the best

players, and also not the winners. The best players should not only be rational themselves, but also

understand that not all the people are rational like them. Hence, the best players should choose a

larger number rather than 1.

4.6   The best players

The best players usually say “I cannot know the correct answer of this problem, before I know

who the other players are. You cannot use the same strategy when you are playing this game with

scientists or with children”. In this experiment, people who provided this kind of reasons finally

gave answers between 15 and 25. There were 4 out of 38 people used this strategy.

The best players use a mixed strategy to optimize their answer. They are rational themselves,

but different from the hyper-rational players. The best players know the theoretical answer is 1,

but they also take account of what levels of rationality other players are. Some players may be

naive, and some players maybe hyper-rational, and also some players maybe thinking about using

a mixed strategy just like them. They know that the answer should be different for different

composition of the population. The best players give the answer based on their estimated

rationality of the population. However, this analysis cannot be done quantitatively without the data.

Simply following what the data shows for the best players’ answers, Uniform [15, 25] is used to

simulate their answer. A more rigorous way of finding an optimal answer like this will be shown

in next section.

4.7   Optimal Strategy and Result Comparison

The following form is a summary of previously mentioned simulation. All these data and

simulation are used to build an optimal strategy.

The following plot visualize the simulation. It involves 1000 groups (38000 numbers in total)

simulated data. Their average is 30.85 (red line on the right in the plot), and 2/3 of the average is

20.56 (red line on the left in the plot), so the answer according to my optimal strategy is 21.

Compare it to the real data collected before, the real average is 30.53, and 2/3 of the average is

20.35, so the winning number should be 20. Even though this is the optimal strategy, the result

still misses the right answer. Actually the simulated answer is pretty close to the winning number,

which indicates the optimal strategy is not bad. However, even the optimal strategy cannot

guarantee a success. This game does need a bit of luck.

The following picture is comparing the simulated result using optimal strategy to a previous

experiment. This is an experiment with 61,139 New York Times readers, so the data is much larger

and more representative. Notice that their final answer is 19, which is not that close to the simulated

answer 21. One reason is that 0 is included as a valid choice in this game, so it is reasonable that

the final answer is a little smaller. The other reason can be that my data involves too many native

players. When doing the simulation, my model generates too many numbers on [80, 95], which

the New York Time readers rarely chose.

5.   Analysis of Variation 1

5.1   Comparing Data to the Original Game

For the first variation, remove the highest 5% of the numbers makes two difference. One is

that it removes some big numbers, which are chosen by some naive players who do not have any

idea about what is happening. Also, because being the highest 5% cannot win the game, people

certainly do not what their answers to be high and get removed. This condition makes players to

choose a smaller number. Both these two conditions lead to a lower average, also a lower final

answer.

The following is the plotted data comparing the data of the original game and the data of the

variation. Most people chose a smaller number than their previous answer, except a few people

who guessed.

5.2   Strategy Analysis

Keep groups the same as the original game. The typical answers of players in each group are

different for the new version.

•   Native players

Usually they are going to guess a number for this variation, just like what they do for

the primary game. I expect these players to choose some number smaller, but the data does

not turn out to be like this. Their answers are random, and the simulation remains the same.

•   Common-rational players I & II

What these players will do is that slightly reduce the number they pick for the original

game. Their answers of the original game are multiplied by 95% to simulate their new

answers. Here the correct way to do it should be calculating the quantile and find the

trimmed mean. However, in practice a player cannot do this calculation without knowing

the real data or using simulation, so the alternative method is to multiply the numbers of

the original game by 95%, which was mentioned by most of the players in their answers.

•   Hyper-rational players

In this game rational players want to choose a smaller number, but hyper-rational

players do not change their answers, because 1 cannot be smaller.

•   The best players

According to the data, the best players reduced their answers to some new numbers

around 90% of their answers of the original game. This shows that they think more

carefully than the common-rational players. They realize that not only the mean is trimmed,

but also players tend to choose smaller numbers. Their answers of the original game are

multiplied by 90% to simulate their new answers

5.3   Optimal Strategy and Result Comparison

The following form and plot is the simulation describes how people choose their answers for

this variation. This result is modified from the original simulation based on players’ new strategies

of this new game.

For this variation, the average is 27.09 (red line on the right in the plot), and 2/3 of it is 18.06

(red line on the left in the plot), so the winning number should be 18. Compare it to the real data

collected before. The real average is 26.92, and 2/3 of it is 17.94, so winning number should be

18. This time the answer by using the optimal strategy and the real winning number are identical.

6.   Analysis of Variation 2

6.1   Comparing Data to the Original Game

For the second variation, players want to choose a bigger number in order to make more money.

However, there is one obstruction. If a choice is too big, it cannot win the game. For instance,

numbers bigger than 67 can hardly win as said before, so players should choose numbers up to 67

even they want to increase their choices to make more money.

The game now becomes choosing a number from 1 to 67. Does this sound familiar? This is

exactly how the hyper-rational players’ strategy starts in the original game. Theoretically, players

can do all the analysis without the condition that the prize is related to the number they choose.

This is because that they have to win first, then start thinking about how much money they can

make given they have already won. Especially for hyper-rational players, this problem and the

original problem are the same. That added condition is a disturbing term.

However, just as how we overturned hyper-rational players’ answers in the original game. Also

in this question, not all the players can realize that the condition is useless in theory. In this

experiment, many of them followed the new condition seriously to choose a bigger number in

order to win more money. Roughly speaking, how much they raised up their answers depends on

how greedy they were.

The following is the plotted data. Most people chose numbers much larger than their previous

answers, some players even choose numbers as big as 100. This is more prominent for naive

players who choose non-rational numbers in the first and the second round.

6.2   Strategy Analysis

•   Native players

Some of them still chose random numbers, but also someone increased their answers a

lot. For those who increased their answer, there were also two different types. One is

increasing their answers above 50. In general, the reason is that the prize in the original

game is 50, and they want to make more money in this variation, so they have to choose a

number at least 50. Another one is increasing their answers to extreme numbers like 100.

These players just naively want to win 100. Therefore, the simulation is separated to three

different uniform distributions as showing in the form in next section.

•   Common-rational players I & II

Usually common-rational players increase their answers by an estimated average

increment. They want to win the game, but also they know that an overlarge number cannot

win. In this experiment, their choices are approximately 1.5 times of their answers of the

original game. The simulation is then adjusted by multiplying parameters with 1.5 .

•   Hyper-rational players

Hyper-rational players should not change their answers, because 1 is still the Nash

Equilibrium of this variation. However, this did not actually happen. There were 3 hyper-

rational players in the original game, and only 1 of them insisted in his answer 1. The other

2 gave up choosing 1. In this case, they were no longer hyper-rational players in this

variation. Their reason was also distinct. They said that they would win at most $1 for

being hyper-rational, so insisting on 1 made them look silly. According to their final

decisions, I put one of them in the common-rational group, and one of them in the native

group, since he was nearly guessing and impatient to tell me his reason.

•   The best players

The best players also increased their answers by an estimated average increment. Generally,

comparing to the common-rational players they are more careful. They have a deeper

insight that a big number will lead to a failure. According to the data, their choices are

approximately 1.2 times of their answers of the original game.

6.3   Optimal Strategy and Result Comparison

The following form and plot is the simulation describes how people choose their answers when

they are doing this variation. It is modified from the original simulation based on players’ new

strategies of this new game. This time not only the numbers they chosen are different, but also the

percentage is different.

For this variation, the average is 50.52 (red line on the right in the plot), and 2/3 of it is 33.68

(red line on the left in the plot), so the winning number should be 34. Compare it to the real data

collected before. The real average is 49.26, and 2/3 of it is 32.84, so the winning number should

be 33. Again, the simulated answer using my optimal strategy is quite close to the winning number,

but it misses the winning number as the original game, which indicates that no matter what the

strategy is, it cannot guarantee a win for this kind of game.

7.   Deficiency

In this study, there are several methodological deficiencies. First, the sample size is fairly small

for only involving 38 players. Also, participants were not randomly selected. In fact, participants

are mainly college students, and many of them are Berkeley students. Therefore, the result is

skewed toward the younger generation. All the simulation distributions are roughly estimated from

the collected data. It is not very accurate. This could be the reason why the optimal answer miss

the real winning number.

8.   Conclusion

This game is more than just picking numbers. It illustrates the common knowledge of

rationality. In other words, what is one’s expectation of other players’ rational levels. It also

highlights the trouble in both natively thinking and thinking too many steps ahead.

A famous real life example of this kind of game is the “Keynesian beauty contest”. It is a contest

to choose the most beautiful face. In this contest, picking the face that others think is the most

beautiful is more likely to win than picking a face that the participant thinks is the most beautiful.

In fact, a player can also pick the face others think that others think that others think - and so on -

is the most beautiful, just like the hyper-rational players in my experiment. However, comparing

to doing the endless others-think inference, start from the composition of the population to

construct a mixed strategy is still optimal.

Reference

David Leonhardt and Kevin Quealy. “Puzzle: Are You Smarter Than 61,139 Other New York

Times Readers?”. 13 August 2015. Web. 19 April 2016.

“Guess 2/3 of the average”, Wikipedia. Web. April 19, 2016

Joshua Hill. Guess the Mean. 2010

“Keynesian beauty contest”, Wikipedia. Web. April 19, 2016

“The Aumann's agreement theorem game (guess 2/3 of the average)”. 09 June 2009. Web. 19,

April 2016.